Newly reducible polynomial iterates

TL;DR

This paper investigates polynomials over fields that become reducible at the nth iterate after being irreducible at the (n-1)th, providing classifications for specific degrees and iterates, and highlighting examples like the golden ratio polynomial.

Contribution

It characterizes when polynomials have newly reducible iterates for certain degrees and iterates, and constructs infinite families of such polynomials over the rationals.

Findings

Existence of infinitely many monic polynomials with newly reducible iterates over $\,\mathbb{Q}$ for specific degree and iterate pairs.

The polynomial $x^2 - x - 1$ has a newly reducible third iterate, exemplifying the phenomenon.

Results include classifications for $(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$ and for degrees $d \equiv 2 \bmod 4$.

Abstract

Given a field $K$ and $n > 1$, we say that a polynomial $f \in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of characterizing, for given $d,n > 1$, fields $K$ such that there exists $f \in K[x]$ of degree $d$ with newly reducible $n$th iterate, and the similar problem for fields admitting infinitely many such $f$. We give results in the cases $(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$ as well as for $(d,2)$ when $d \equiv 2 \bmod{4}$. In particular, we show that for all these $(d,n)$ pairs, there are infinitely many monic $f \in \mathbb{Z}[x]$ of degree $d$ with newly reducible $n$th iterate over $\mathbb{Q}$. Curiously, the minimal polynomial $x^2 - x - 1$ of the golden ratio is one example of $f \in \mathbb{Z}[x]$ with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.